LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 3
a natural way to an explicit solutions procedure by factorization for a class of invariant
Hamiltonian flows, as in (1.5) above. For the convenience of the reader we present a brief
summary of the relevant results of classical J^-matrix theory in the Appendix.
The main task of this paper is to give a Lie-algebraic interpretation of the results in
[MV]. The underlying algebra turns out to be a loop algebra with an associated classical
.R-matrix which is an appropriate generalization of the i?-matrix arising in the dynamical
theory of the Cholesky algorithm, as described in [DLT]. The discrete systems of Moser
and Veselov are time-one maps of integrable Hamiltonian systems, and the solution pro-
cedure (1.7), (1.8), and its analogs for all the systems considered in [MV], is precisely the
factorization procedure, here of Riemann-Hilbert type, suggested by the general theory of
classical J?-matrices. A single loop algebra suffices to describe all the systems in [MV]: all
that differs from one system to the next, is the particular choice of coadjoint orbit of the
associated loop group (however, see §3).
In order to describe our results in greater detail, we need more information from [MV].
In the case (1.4),
S = Y,
XkJXk+i - (1-9)
the Euler-Lagrange equations take the form
X^J + Xk-tJ^AkXk (1.10)
where Ajt = A^ is a matrix Lagrange multipler: A* is uniquely determined by Xk-i, Xk,
Xfc+i, but not uniquely determined by A~jb_i, A"*. Thus the discrete Euler-Lagrange equa-
tions lead in general to a correspondence (Xk-\'Xk) | —• (Xk,Xk+i)i and the choice of a
particular mapping ^ is equivalent to the choice of a particular branch of the correspon-
uk = xZxk-1 eO(N), (l.ii)
and using A& = Ajf, equation (1.10) can be rewritten as a "discrete Euler-Arnold equation"